11.6: Conditional Probability

Introduction

Bike Shop Repairs

On Thursday, Carey was in charge of answering the phones and booking appointments for bike repairs. The bike shop repairs bikes on Monday, Tuesday and Wednesday mornings and on Thursday and Friday afternoons. All appointments are booked randomly. The person making the appointment can choose or the person answering the phone can choose.

Carey booked two appointments right away.

What are the chances that both of the these appointments were booked on a Monday, Tuesday or Wednesday morning?

Answering this question will require you to understand conditional probability. This lesson will teach you all that you need to know so that you will be able to figure out the solution to the problem by the end of the lesson.

What You Will Learn

By the end of this lesson, you will be able to complete the following skills.

Recognize and distinguish among independent and dependent events.

Recognize and apply the definition of conditional probability to find probabilities in finite sample spaces.

Calculate the probabilities of a series of independent events, mutually exclusive events and events that are not mutually exclusive.

Make predictions involving conditional probability.

Teaching Time

I. Recognize and Distinguish among Independent and Dependent Events

In the last few sections, you have been learning all about probability. Now we can think about different events and how these events impact each other. Take a look at this example.

Suppose you have two events:

Event A: Toss 5 on the number cube

Event B: Spin blue on the spinner

The probability of each of these events by itself is easy enough to compute. In general:

Clearly, the first event affected the outcome of the second event in this situation. So the two events are NOT independent. In other words, they are dependent events.

Definition: If the outcome of one event has an effect on the outcome of a second event, then the two events are dependent events.

II. Recognize and Apply the Definition of Conditional Probability to Find Probabilities in Finite Sample Spaces

Sometimes the outcome that you get when figuring out a probability is what we call “conditional.” This means that we will only an outcome if the conditions designed to cause a specific result. Let’s look at an example.

Example

Consider a jar with 4 black marbles and 6 white marbles. If you pull out 2 marbles from the jar randomly, one at a time, without replacing the first marble, what is the probability that both marbles will be white?

Start by approaching the problem the same as you would with independent events. The probability of the first marble being white is:

The same general method works for calculating any two (or more) dependent events.

Example

Jack’s Catering Service is accepting weekday appointments for Monday through Thursday, and weekend appointments for Friday through Sunday. If appointment dates are made randomly, what is the probability that 2 weekdays will be the first 2 days to be booked?

Conditional probabilityinvolves situations in which you determine the probability of an event based on another event having occurred.

For example, suppose you roll two number cubes on a table. The first cube lands face up on 5. The second cube falls off of the table so you can’t see how it landed. Given what you know so far, what is the probability that the sum of the number cubes will be 9?

To solve this problem, consider the entire sample space for rolling two number cubes.

Notice that we write the conditional probability as \begin{align*}P (9 | 4)\end{align*}. You can read this as:

\begin{align*}P (9|4) \Longleftarrow\end{align*} the probability of 9, given 4

Here are some other examples of how to read this notation.

\begin{align*}P (B|A) \Longleftarrow\end{align*} the probability of \begin{align*}B\end{align*}, given \begin{align*}A\end{align*}

\begin{align*}P (7|3) \Longleftarrow\end{align*} the probability of 7, given 3

\begin{align*}P (\text{heads}| \text{tails}) \Longleftarrow\end{align*} the probability of heads, given tails

\begin{align*}P (\text{red}| \text{blue}) \Longleftarrow\end{align*} the probability of red, given blue

The probability is determined because certain factors are in place.

III. Calculate Probabilities of a Series of Independent Events, Mutually Exclusive Events and Events that are not Mutually Exclusive

Sometimes, we have mutually exclusive events and we have events that overlap and are not mutually exclusive.

In a previous section, you saw that event \begin{align*}R(\text{red})\end{align*} and event \begin{align*}T(\text{top})\end{align*} are overlapping events because both events share one outcome – red-top. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.

To calculate the probability of overlapping events, list the sample space and find the favorable events.

Think back for a minute and remember what we mean when we talk about conditional probability.

Conditional probabilityinvolves situations in which you determine the probability of an event based on another event having occurred.

We can use conditional probability to determine probabilities, but also to make predictions. Look at this example.

Example

A stack of 12 cards has the Ace, King, and Queen of all 4 suits, spades, hearts, diamonds, and clubs. What is the probability that if you draw 2 cards randomly, they will both be hearts? Make a prediction.

Step 1: Draw the first card. The probability of it being a heart is 3 of 12.

Step 2: Now draw the second card. Since the first card was a heart, there are only 11 cards left and only 2 of them are hearts.

Step 3: Calculate the final probability.

So \begin{align*}P(\text{heart and heart}) = \frac{1}{22}\end{align*}. You would predict that both cards would be hearts \begin{align*}\frac{1}{22}\end{align*} of the time.

Real-Life Example Completed

Bike Shop Repairs

Here is the original problem from the introduction. Reread it and then solve it for the correct probability.

On Thursday, Carey was in charge of answering the phones and booking appointments for bike repairs. The bike shop repairs bikes on Monday, Tuesday and Wednesday mornings and on Thursday and Friday afternoons. All appointments are booked randomly. The person making the appointment can choose or the person answering the phone can choose.

Carey booked two appointments right away.

What are the chances that both of the these appointments were booked on a Monday, Tuesday or Wednesday morning?

Now answer the question at the end of the problem.

Solution to Real – Life Example

To work on this probability, first we must determine the probability of the first appointment booked being on a Monday, Tuesday or Wednesday. There are five possible days for appointments, but three favorable outcomes.

Probability of first appointment being Mon, Tues or Weds \begin{align*}=\frac{3}{5}\end{align*}

Probability of second appointment being Mon, Tues or Weds \begin{align*}=\frac{2}{4}\end{align*} or \begin{align*}\frac{1}{2}\end{align*}